Tuesday, June 02, 2015

Fun with formality

T/F: Romer thinks that economists should not try to use the mathematics of Debreu/Bourbaki and should instead use math in the less formal way that physicists and engineers use it...

[False.]

[H]and-waving and verbal evasion...is the exact opposite of the precision in reasoning and communication exemplified by Debreu/Bourbaki, and I’m for precision and clarity.

But that comment got me thinking about formalism in econ math, and I thought I'd share some thoughts.

I've never actually read any Bourbaki papers, but Bourbaki was a club of mostly French mathematicians who got together in the 1930s and insisted that mathematicians be very formal. They got their wish, and the result is the rigorous, formalistic style of modern math papers. But physicists and engineers never followed this convention, preferring to derive essential results and let mathematicians pick up after them by putting in the formality.

There are economists who follow both conventions. You see some papers that use a very formal, terse, axiom-theorem-proof style similar to what you'd see in a mathematics journal. And you see some papers that use a more informal, "here's an equation that I think describes something" methodology that you might see in an engineering journal.

An example of the formal style would be "The Simple Theory of Temptation and Self-Control," by Farak Gul and Wolfgang Pesendorfer. This paper introduces a wrinkle to the standard classical theory of intertemporal consumer decision-making - they allow people to have preferences over the sets of choices they are given, such as when people on a diet might not want to see the dessert menu. This wrinkle is inserted in the form of a new "axiom", called the Axiom of Set Betweenness. The presentation in the paper tends to look like this:

An example of the informal style would be "Golden Eggs and Hyperbolic Discounting," by David Laibson. This paper also introduces a wrinkle to the classical theory of intertemporal consumer decision-making. The wrinkle is a new functional form for the consumer's discount function. The presentation in the paper tends to look like this:

In fact, both of these papers are motivated by some of the same empirical phenomena. But they go about it in very different ways. Gul and Pesendorfer introduce an entirely new framework, while Laibson tweaks a functional form. In other words, Gul and Pesendorfer rewrite all of the rules for how decision-making is thought to operate, while Laibson sticks in something that works. As a result, it's natural for Gul and Pesendorfer to use a very formal framework, since formal things are more general and can build a foundation for many other theorists to work with. Laibson doesn't have any need to be so formal.

I imagine that some people complain about the formalism in Gul and Pesendorfer because it's hard for them to read. But after you learn to speak that language, it's actually easy to read - in many ways, easier than English. Formal math language forces you to read like a computer, which means you don't miss anything, while English tempts you (heh) to gloss over important parts as you scan through paragraphs.

In general, I think formal math style is no worse or better than informal engineering style. It's just a matter of personal preference.

Another thing that might annoy people about Gul and Pesendorfer's formalism is the clunkiness of doing economics this way. Do we really want to have to re-axiomatize all of consumer decision-making every time we see people doing something weird? Isn't the overhead of formalism a big waste of time and effort?

Well, maybe. If we take the Laibson paper seriously, all we have to do is to introduce a hyperbolic discounting function whenever we suspect it might make a difference in a model. That's equivalent to just setting the parameters of the hyperbolic discounting function to approximate a classic, non-hyperbolic discount function whenever we don't think it's interesting. But if we take the Gul and Pesendorfer paper seriously, we might have to reformulate all our theories. It's just not clear when the Axiom of Set Betweenness might apply. An axiom is just a lot more general than a parametrization. It seems to me that that's what you can lose from formalism - a clear sense of when the new stuff might make a difference.

But in the end, I bet that people use the Gul & Pesendorfer stuff in the exact same way they use the Laibson stuff - they apply it when they think it might make a difference, and forget about it at all other times. So formalism vs. informalism again just comes down to a matter of personal preference.

26 comments:

But it does seem weird ... if you read the first segment the word "tempting" appears. It is jarringly out-of-place. It is like coding in assembly with registers and bit shifts and coming across a method for a dynamically typed object. Such a high-level concept shouldn't appear in low-level code.

I think this is related to the "physics envy" charge -- physics should be written in assembly. Economics should be written in python (or R or apparently Mathematica more often than I thought).

You're free to use whatever formalism is fun for you -- just be ready for people to say it's strange.

How does the analogy work? Coding in Python vs Assembly is a choice fundamentally on how quickly you can throw up some code vs how quickly you want that code executed. You cant throw up something in Python and then hide the fact that it doesnt work the way certain people masquerading as Economists who look tremendously like male cologne salesmen named Niall.

In Noah is saying that the different styles really are about whether someone feels comfortable enough expressing themselves mostly in English or logic.

And since cultural context matters (and now that Noah has a tenure track position and cant spit hot lead like he used to when he was a rage filled PhD student but grounded with dignity and therefore couldnt just go to that anonymous right wing circle jerk of male insecurity known as EJMR to do so but instead did on here) old school Noah probably would have added:

the further implication is that the guy writing in English is less autistic and a generally better human being while the guy writing in logic was too much of a punk bitch to continue studying physics or computer science or math and retreated to the liberal arts of math -- economics --. There he chiefly concentrates on producing reams of useless paper while his more courageous intellectual peers are out there writing new code or building new polymers or generally being useful members of civilization. When this sad pantomime of humanity isnt writing papers 2 foreign grad students are going to be forcefed by a cruel supervisor, he is day dreaming about Tyler Cowen-ing some rich oligarch into funding a center of mediocrity so he can too try to hide his autism better by appropriate some worldliness by recommending subpar dining and entertaining options for a blog audience just on this side of zerohedge's neo-nazis men's rights brigade of mendacity.

I don't think the choice is completely one dimensional, but I do get your point. I think a better version of the analogy uses Turning machines instead of assembly. I saw coding in assembly closer to programming/proving theorems about a Turning machine -- that is to say 'fundamental' research vs engineering.

The difficulty is that adding concepts like "temptation" to a Turing machine crosses several levels of computing research -- much more an AI concept.

Ah, Noah, let a thousand (or maybe at least two) flowers bloom! How nice, not that I strongly disagree with your main argument here.

As it is, reading Romer's second round version of his mathiness argument, he pounds the table on how he is not against high math, inclucing of the Bourbaki variety. Supposedly his complaint is about people verbalizing about stuff in their papers that is not there in the math, which presumably none of us should approve of, whatever our views of various degrees or levels of math that are appropriate, although some others, notably Andolofatto. who claim that his complaint about the 2009 Lucas paper does not hold.

I would note that previously I tried to post a comment that never made it (will this?) to the point that it looked like Romer, and certainly some of the rest of us, may sometimes object that some of the supposedly sophisticated math being used in certain models, most notoriously in DSGE ones, is in fact insufficient and too simple-minded for what is involved, particularly when there may be nonlinearities and associated multiple equilibria (a point that Roger Farmer makes, although not generally complaining about the math being too low level). For me this is where "mathiness" becomes especially annoying; people strutting about bragging about how rigorous their models supposedly are when they are full of ridiculously simplifying and simple-minded assumptions that allow them to get nice neat but nearly useless results. It is one thing to brag about how one is assuming general equilbirium, and it is another to blatantly ignore a lot of issues any respectable micro GE theorist would raise about the nature of one's model.

Just out of curiosity, would you name a couple. In the post, you named 2 economists who follow one convention (full mathematical rigor), and one who follows the other (engineering/physics), but none who follows both.

Romer's Bourbaki reference suggests to me a lack of understanding of a very fundamental distinction between mathematics and science (and engineering).

Bourbaki was an attempt to axiomatize all of mathematics starting from the Zermelo-Fraenkel axiom system of set theory. Earlier Russell and Whitehead had a similar aim starting from their system in Principia Mathematica.

The central issue is that Bourbaki or PM treats mathematics as a purely formal system . A purely formal system involves the manipulation of meaningless symbols according to fixed rules. Or as Bertrand Russell famously defined it:

" Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

The key point is that the symbols, formulae, axioms etc in a purely formal system have no empirical content. The symbols are undefined and the formulae and axioms are defined in terms of the meaningless symbols.

A scientific or engineering model is very different. Here the terms in your model cannot be meaningless. There needs to be an unambiguous mapping from terms in your formal model to observable and measurable entities in the real world domain whose behaviour you are trying to model.

Bourbaki was about purely formal systems. Scientists and engineers do not use such a purely deductive axiomatic approach because it does not work in formal models about real world phenomena.

Anyone who refers to Bourbaki in the context of economics is effectively saying (with apologies to Bertrand Russell) :

Thus *economics* may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Pure mathematics or metamathematics is not my field and if I've got any thing wrong, perhaps a mathematician can correct me. Prof Barkley Rosser's father was an important player in some of these debates, and I'm sure he knows more about these issues than most I or most economists do.

I partially agree. A few areas of economics - pure game theory and some decision theory - are really just pure math that gets done in an econ department. There is no need for the "agents" or "decision makers" in many of those "theories" to have any correspondence with real agents or decision makers, since the question is simply about the math of how certain idealized objects behave, not about reality. (Note that this could still be of practical use, since we might design computer algorithms based on these findings.)

But Gul and Pesendorfer's paper purports to be about reality, so they should really call their axioms "postulates". Not that I'm particularly up in arms about that...in the end, it ends up getting applied in the same way as less formal formulations like Laibson's.

H, I do not disagree with your presentation of the matter. However, Noah is also right that some of those playing the Bourbaki game in economics sometimes kid themselves into thinking that they are saying something that may have direct relevance to the real world. An interesting case, and also an answer to the query by marcel proust, involves not game theory, but the existence proofs of competitive general equilibrium, especially that by Arro and Debreu. Debreu in particular was a top student of Cartan, probably the ultimate leader of the Bourbaki group (the name Bourbaki came from a French general of an earlier period who was not a mathematician). Debreu was effectively a true believer in Bourbakism, and probablby its strongest, or at least moet inflluential advocate in econoics, with his Nobel Prize address maybe his most publicized and articulate statement of his case. But, for all his pure Bourbakism, I think he thought his work on these matters had some relevance to the real world, even if he may never have said that to Cartan. Arrow certainly played the Bourbakist game with Debreu and on several other occasions, fully capable of doing pure theory along axiomatic lines. However, he has also written extensivelhy on policy quesions, where he has followed more the engineering approach, to use Noah's distinction here, focusing on particular equations and their use to explain or analyze policy issues in real economies of various sorts, without explicitly deriving these equations from a fundamental axiomatic foundation, although on more than one occasion he has made reference to how such equations might be derived from such foundations and how they might fit into a broader general equilibrium framework. In short, Arrow has certainly gone both ways, and I suspect does not get all that worked up about this distinction, which I think his late coauthor took far more seriously.

How are you defining formalism v. the engineering approac?. Maybe this is a bit ironic given the subject matter, but I'm a little foggy on your definitions - and whether you think they extend beyond, say, rigorously vs. informally constructing utility functions.

I think what Romer was saying is that the choice between these two styles has nothing whatsoever to do with his attack on mathiness. He's not taking a position for or against one or the other.

I don't know why he bothered. Did somebody somewhere think his "mathiness" had something to do with the difference between these two styles? Really? I guess he was just imagining an extreme misunderstanding for illustration purposes.

Anyway, I do think Romer is onto something. It is common for economists to hide poorly thought through ideas behind impressive-looking math. But even his coding illustration doesn't fully clarify for me what he means by mathiness. His examples still seem a grab-bag of flaws.

It also seems to me there are two different lemon markets at work. One is within econ journals, and has to do with the very narrow scrutiny received and the tendency of publication to be decided by fellow members of the cliques that Romer describes. The scrutiny could be improved I think if all econ papers were forced to fully explain their ideas in text and math. I understand that most of them will only be read by an in-group who know the calculus and the whole tradition behind the formula and might feel their time is being wasted by having to read the definitions of terms as if this were the first time they were seeing this kind of formula. But it would greatly widen the readership and increase the scrutiny. Having to write more prose alongside the math would I think also force econ-paper writers to think through their ideas more fully. Math is good for precision of communication, but says nothing about real-world plausibility.

The other kind of econ-math lemons market is in popular economics, such as Piketty's claim that if the net growth rate halves the wealth stock doubles. This has more to do with deceiving lay readers by exploiting the seeming certainty of math while not mentioning how you've limited the model.

It is not common to hide poor ideas behind math, because it doesn't fool the referees. Just because you think calculus is "impressive" and "difficult" doesn't mean economists do. You are clearly not well educated, and should probably refrain from public speaking until you correct that situation.

No, I mean integral and differential calculus, but the first writer missed my argument entirely. Part of which was that Romer is describing, especially re: the paper with the "location" issue, how bad papers get published because referees are too much of an in-group and don't provide real scrutiny. And I'm arguing that demanding fuller prose explanations in addition to math would open up economics journals to broader scrutiny. I also think that while writing in math forces precision, writing in prose forces the writer to think through sense and realism. I never suggested that using math hides anything from the referees, you silly goose.

As a mathematician trained in pure math and now collaborating with physicists & others, I have a different take on some of these issues. First Bourbaki. "He" didn't advocate a higher standard of rigor, just a more formal style of exposition. Your examples of economic exposition are similar. In both cases someone presents a mathematical model, arguing that certain equations or other mathematical structures capture some aspect of (economic) reality. They are doing the same kind of thing with different expository style.

The pure mathematician/applied physicist difference concerns reasoning once the basic model is established. There is the mathematicians' completely airtight deduction and the physicists' "it probably goes like this". An extreme example is the physicist Parisi, who reasoned about the symmetry of something in n dimensions, only to take the limit n --> 0. He states that a zero dimensional space must be very complicated. Some of what mathematicians call the "Parisi conjectures" have rigorous proofs, others not yet (or ever?).

Physicists, and the rest of the technical world, are willing to take basic qualitative facts on faith. For example, they are willing to do perturbation expansions without first proving that the quantity is differentiable with respect to the perturbation parameter. In my experience, economists waste a lot of time proving things that physicists would, rightly, take for granted. And economists aren't that good at proofs. They often get them wrong (mangle hypotheses, etc.). The appearance of rigor, imitating the Bourbaki writing style, does not make it correct.

If the quantity isn't differentiable with respect to the perturbation parameter,(deep breath),

then the quantity probably has a *cusp* or a *discontinuity*. You can still do a perturbation, but you have to walk through the discontinuities if you want to *get the right result*. Doing the expansion without proving that the quantity is differentiable can give blatantly wrong results if there is such a cusp or discontinuity.

However! In physics, the cusps and discontinuities are often *known in advance from experiments*, and referred to by names such as "phase change". So there's that.

lgm brings up an interesting point that I do not think has ever happened in economics, namely people using a mathematical concept that has not really been approved of or shown to be fully consistent with standard ZFC founded math. Parisi conjectures are an example, and the Dirac delta function and related Heaviside is another. Was a long time before the delta function was viewed as really properly mathematically acceptable, but physicists went around using it and using it because it was mighty handy for solving some serious problems.